A note on differentials of holomorphic functions

TL;DR

This paper provides an intrinsic characterization of holomorphic Lipschitz functions on Banach space balls and offers explicit methods for identifying such functions when the space has a Schauder basis.

Contribution

It offers a new intrinsic characterization of the subspace of holomorphic Lipschitz functions and explicit criteria for membership in this subspace for Banach spaces with a Schauder basis.

Findings

Characterization of holomorphic Lipschitz functions on Banach space balls.

Explicit criteria for membership in the subspace when a Schauder basis exists.

Clarification of the structure of these function spaces in infinite-dimensional settings.

Abstract

Recently, in arXiv:2304.07149, a bridge was made between the very active area of spaces of Lipschitz real functions on a metric space and holomorphic functions on an open subset of a Banach space. This was done by introducing and studying the space $\mathcal HL_0(B_X)$ of holomorphic Lipschitz functions defined on $B_X$, the open unit ball of the complex Banach space $X$ vanishing at 0. There it was proved that this space is isometrically isomorphic to a subspace of $\mathcal H^\infty(B_X, X^*)$, the space of bounded holomorphic mapping with values in the topological dual of $X$. In that paper it was shown that this subspace was a proper one, except in the one dimensional case. The goal of this note is to give an intrinsic characterization of the elements of that subspace. Moreover, in the case where $X$ additionally has a Schauder basis, it is shown that there is an explicit way to calculate whether and element of $\mathcal H^\infty(B_X, X^*)$ belongs or not to that subspace.